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Herausgeber: John Wiley & Sons
Kategorie: Wissenschaft und neue Technologien
Serie: Wiley Series in Probability and Statistics
Sprache: Englisch

Beschreibung

Probability and Conditional Expectations bridges the gap between books on probability theory and statistics by providing the probabilistic concepts estimated and tested in analysis of variance, regression analysis, factor analysis, structural equation modeling, hierarchical linear models and analysis of qualitative data. The authors emphasize the theory of conditional expectations that is also fundamental to conditional independence and conditional distributions.

Probability and Conditional Expectations

Presents a rigorous and detailed mathematical treatment of probability theory focusing on concepts that are fundamental to understand what we are estimating in applied statistics.
Explores the basics of random variables along with extensive coverage of measurable functions and integration.
Extensively treats conditional expectations also with respect to a conditional probability measure and the concept of conditional effect functions, which are crucial in the analysis of causal effects.
Is illustrated throughout with simple examples, numerous exercises and detailed solutions.
Provides website links to further resources including videos of courses delivered by the authors as well as R code exercises to help illustrate the theory presented throughout the book.

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Veröffentlichungsjahr: 2017

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Leseprobe

WILEY SERIES IN PROBABILITY AND STATISTICS

Established by Walter A. Shewhart and Samuel S. Wilks

Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Geof H. Givens, Harvey Goldstein, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay

Editors Emeriti: J. Stuart Hunter, Iain M. Johnstone, Joseph B. Kadane, Jozef L. Teugels

The Wiley Series in Probability and Statistics is well established and authoritative. It covers many topics of current research interest in both pure and applied statistics and probability theory. Written by leading statisticians and institutions, the titles span both state-of-the-art developments in the field and classical methods.

Reflecting the wide range of current research in statistics, the series encompasses applied, methodological and theoretical statistics, ranging from applications and new techniques made possible by advances in computerized practice to rigorous treatment of theoretical approaches. This series provides essential and invaluable reading for all statisticians, whether in academia, industry, government, or research.

A complete list of titles in this series can be found at http://www.wiley.com/go/wsps

Probability andConditional Expectation

Fundamentals for the Empirical Sciences

Rolf Steyer

Institute of Psychology, University of Jena, Germany

Werner Nagel

Institute of Mathematics, University of Jena, Germany

Registered officeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the authors to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

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Library of Congress Cataloging-in-Publication Data

Names: Steyer, Rolf, 1950– | Nagel, Werner, 1952–. Title: Probability and conditional expectation : fundamentals for the empirical sciences / Rolf Steyer, Werner Nagel. Description: Chichester, West Sussex : John Wiley & Sons, Inc., 2017. | Includes index. Identifiers: LCCN 2016025874 | ISBN 9781119243526 (cloth) | ISBN 9781119243489 (epub) | ISBN 9781119243502 (epdf) Subjects: LCSH: Conditional expectations (Mathematics) | Random variables. | Independence (Mathematics) | Dependence (Statistics) | Measure theory. | Probability and statistics. | Correlation (Statistics) | Multivariate analysis. | Regression analysis. | Logistic regression analysis. | Measure algebras. | Probabilities. Classification: LCC QA273 .S75325 2017 | DDC 519.2–dc23 LC record available at https://lccn.loc.gov/2016025874

A catalogue record for this book is available from the British Library.

For my wonderful wife and children.—RST

Preface

Why another book on probability?

What is it about?

For whom is it?

Prerequisites

Acknowledgements

About the companion website

Part I: Measure-theoretical foundations of probability theory

1: Measure

1.1 Introductory examples

1.2 σ-Algebra and measurable space

1.3 Measure and measure space

1.4 Specific measures

1.5 Continuity of a measure

1.6 Specifying a measure via a generating system

1.7 σ-Algebra that is trivial with respect to a measure

1.8 Proofs

Exercises

Solutions

2: Measurable mapping

2.1 Image and inverse image

2.2 Introductory examples

2.3 Measurable mapping

2.4 Theorems on measurable mappings

2.5 Equivalence of two mappings with respect to a measure

2.6 Image measure

2.7 Proofs

Exercises

Solutions

3: Integral

3.1 Definition

3.2 Properties

3.3 Lebesgue and Riemann integral

3.4 Density

3.5 Absolute continuity and the Radon-Nikodym theorem

3.6 Integral with respect to a product measure

3.7 Proofs

Exercises

Solutions

Part II: Probability, random variable, and its distribution

4: Probability measure

4.1 Probability measure and probability space

4.2 Conditional probability

4.3 Independence

4.4 Conditional independence given an event

4.5 Proofs

Exercises

Solutions

5: Random variable, distribution, density, and distribution function

5.1 Random variable and its distribution

5.2 Equivalence of two random variables with respect to a probability measure

5.3 Multivariate random variable

5.4 Independence of random variables

5.5 Probability function of a discrete random variable

5.6 Probability density with respect to a measure

5.7 Uni- or multivariate real-valued random variable

5.8 Proofs

Exercises

Solutions

6: Expectation, variance, and other moments

6.1 Expectation

6.2 Moments, variance, and standard deviation

6.3 Proofs

Exercises

Solutions

7: Linear quasi-regression, covariance, and correlation

7.1 Linear quasi-regression

7.2 Covariance

7.3 Correlation

7.4 Expectation vector and covariance matrix

7.5 Multiple linear quasi-regression

7.6 Proofs

Exercises

Solutions

8: Some distributions

8.1 Some distributions of discrete random variables

8.2 Some distributions of continuous random variables

8.3 Proofs

Exercises

Solutions

Part III: Conditional expectation and regression

9: Conditional expectation value and discrete conditional expectation

9.1 Conditional expectation value

9.2 Transformation theorem

9.3 Other properties

9.4 Discrete conditional expectation

9.5 Discrete regression

9.6 Examples

9.7 Proofs

Exercises

Solutions

10: Conditional expectation

10.1 Assumptions and definitions

10.2 Existence and uniqueness

10.3 Rules of computation and other properties

10.4 Factorization, regression, and conditional expectation value

10.5 Characterizing a conditional expectation by the joint distribution

10.6 Conditional mean independence

10.7 Proofs

Exercises

Solutions

11: Residual, conditional variance, and conditional covariance

11.1 Residual with respect to a conditional expectation

11.2 Coefficient of determination and multiple correlation

11.3 Conditional variance and covariance given a σ-algebra

11.4 Conditional variance and covariance given a value of a random variable

11.5 Properties of conditional variances and covariances

11.6 Partial correlation

11.7 Proofs

Exercises

Solutions

12: Linear regression

12.1 Basic ideas

12.2 Assumptions and definitions

12.3 Examples

12.4 Linear quasi-regression

12.5 Uniqueness and identification of regression coefficients

12.6 Linear regression

12.7 Parameterizations of a discrete conditional expectation

12.8 Invariance of regression coefficients

12.9 Proofs

Exercises

Solutions

13: Linear logistic regression

13.1 Logit transformation of a conditional probability

13.2 Linear logistic parameterization

13.3 A parameterization of a discrete conditional probability

13.4 Identification of coefficients of a linear logistic parameterization

13.5 Linear logistic regression and linear logit regression

13.6 Proofs

Exercises

Solutions

14: Conditional expectation with respect to a conditional-probability measure

14.1 Introductory examples

14.2 Assumptions and definitions

14.3 Properties

14.4 Partial conditional expectation

14.5 Factorization

14.6 Uniqueness

14.7 Conditional mean independence with respect to

14.8 Proofs

Exercises

Solutions

15: Effect functions of a discrete regressor

15.1 Assumptions and definitions

15.2 Intercept function and effect functions

15.3 Implications of independence of

and

for regression coefficients

15.4 Adjusted effect functions

15.5 Logit effect functions

15.6 Implications of independence of

and

for the logit regression coefficients

15.7 Proofs

Exercises

Solutions

Part IV: Conditional independence and conditional distribution

16: Conditional independence

16.1 Assumptions and definitions

16.2 Properties

16.3 Conditional independence and conditional mean independence

16.4 Families of events

16.5 Families of set systems

16.6 Families of random variables

16.7 Proofs

Exercises

Solutions

17: Conditional distribution

17.1 Conditional distribution given a σ-algebra or a random variable

17.2 Conditional distribution given a value of a random variable

17.3 Existence and uniqueness

17.4 Conditional-probability measure given a value of a random variable

17.5 Decomposing the joint distribution of random variables

17.6 Conditional independence and conditional distributions

17.7 Expectations with respect to a conditional distribution

17.8 Conditional distribution function and probability density

17.9 Conditional distribution and Radon-Nikodym density

17.10 Proofs

Exercises

Solutions

References

List of Symbols

Author index

Subject index

EULA

List of Tables

Chapter 2

Table 2.1

Table 2.2

Chapter 4

Table 4.1

Chapter 5

Table 5.1

Table 5.2

Chapter 9

Table 9.1

Table 9.2

Chapter 10

Table 10.1

Chapter 11

Table 11.1

Table 11.2

Chapter 12

Table 12.1

Chapter 13

Table 13.1

Chapter 14

Table 14.1

Table 14.2

Table 14.3

Chapter 15

Table 15.1

Table 15.2

Chapter 16

Table 16.1

Table 16.2

Chapter 17

Table 17.1

Guide

Cover

Table of Contents

Preface

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Acknowledgements

This book could not have been written without the help of many. First of all, we thank Ivailo Partchev, who prepared the LaTeX framework and many of the figures, tables, and boxes. Some of the figures have been produced by Désirée Thielemann and Julie Toussaint, who also cared for references, read some of the chapters, and hinted at errors. For supporting us with respect to LaTeX, finding errors, or suggesting other improvements, we also thank Karoline Bading, Marcel Bauer, Sonja Hahn, Gregor Kappler, Christoph Kiefer, Andreas Neudecker, Axel Mayer, Erik Sengewald, Jan Plötner, Carolin Rebekka Scheifele, and Tom Landes. Thanks are also due to Ernesto San Martin for suggesting section 1.7 and proposition (iv) of Theorem 16.37. The proof of Lemma 12.38 is due to Peter Vogel. Finally, we would like to thank our students who kept us thinking on how to improve the text.